Spread 3 tablespoons butter over one side of each slice of bread, then cut bread into 1" pieces. Scatter half of bread evenly into prepared dish and sprinkle with half of mozzarella, Parmesan, and tomatoes. Repeat with remaining bread, mozzarell, Parmesan, and tomatoes.
Whisk together eggs, milk, onion, salt, pepper, mustard, and Tabasco and pour evenly over bread. Press lightly on the bread to submerge. COver tightly with plastic wrap and refrigerate at least 8 hours and up to 36 hours.
Remove dish from refrigerator and uncover. Preheat oven to 350 degrees. Adjust oven rack ti middle position and heat oven to 350 degrees. Bake until puffed and golden brown, about 1 hour. Serve immediately.